On The Equilibrium Of Continuous-time Insider Trading Based On Characteristics Of Different Agents

Insider trading is that those market participates,called insiders,pos-sess partial or complete information on a risk asset,and take advantage of this information to maximize their profits under the cover of noisy trades,while market makers will set a reasonable price of the risk asset to guarantee market efficiency on the basis of their market information.However,in a real financial market,market agents including insiders,market makers and noisy traders will have different motives,hold differ-ent market information,or owns inconsistent risk appetites.So how to play the game among these agents to obtain a market equilibrium is a hot topic in financial research field.In this thesis,based on characteristics of different agents,we will investigate some practical mathematical models of continuous-time insider trading,and analyze the corresponding market equilibria and their properties.Furthermore,theoretically,some related mathematical problems are also studied.The major contributions can be summarized as follows.Firstly,based on characteristics of risk preference of insiders,i.e.,risk neutral or risk averse,we propose corresponding continuous-time in-sider trading models with a dynamic risk asset.By the use of filtering theory and principle of stochastic dynamic programming,we establish the existence and uniqueness of a linear Bayesian market equilibrium,consisting of optimal trading strategy and semi-efficient pricing rule.At last,some impacts of risk-aversion degree or risk-asset volatility on e-quilibrium characteristics are analyzed.Both theoretical analysis and simulations show that in the equilibrium,as time goes by,whether the insider is risk neutral or risk averse,the trading intensity is increasing while the residual information is decreasing;however,the price pressure with risk-aversion is decreasing while the price pressure with risk-neutral is constant;and when the insider becomes less and less risk-averse,the risk-averse equilibrium converges to the risk-neutral one.Secondly,based on characteristics of partial observation of market makers on a risk asset,we propose some corresponding continuous-time insider trading models with random deadline.By combining stochastic dynamic programming principle,maximum principle with filtering theo-ry,we establish both the existence and uniqueness of a linear Bayesian market equilibrium and that of a linear expect market equilibrium respec-tively.Then some similarities and differences between these two kinds of equilibria,and some impacts of partial observation of market makers or random deadline on equilibrium characteristics are analyzed.Both the-oretical analysis and simulations show that in the equilibrium,the less accurate information observed by market makers,the lower the market liquidity and the trading intensity are,while both the expected profit of insider and the residual information are increasing,and as time goes by,both the intensity and the market depth are increasing while the residual information is decreasing;the longer the mean of random time,the more the residual information is but the weaker the trading intensity is,how-ever,there is a threshold of trading time,half of the mean of the random time,such that if and only if after it the market liquidity is increasing with the mean of random time increasing.Thirdly,based on characteristics of memory of noisy traders,we pro-pose a corresponding continuous-time insider trading model with random deadline.By the use of fractional Brownian filtering theory and max-imum principle,we give some necessary condition of the existence of a linear expect market equilibrium,and find that under some appropriate conditions,as the noise traders'memories are weaker and weaker,both the optimal trading intensity and the residual information converge to those when noise traders without any memory,respectively.In addition,with the help of fractional filtering theory and the notion of reference probability space,we establish the existence and uniqueness of Q₀-weak solution to some kind of conditional mean field stochastic differential equation driven by fractional Brownian motion.These results above will provide theoretical guidance and reference both for insiders to choose optimal strategies and for market makers to set price efficiently in insider trading markets.